It's not a fun job, but heating with soldering iron and then unscrewing is the way to go.

Have you tried the standard Roughing the nipple/tit of the socket and moving the wire there? You mentioned soldering the sockets I didn't know if that's what you meant?

Use a hotter soldering iron? Cutting and soldering the holder instead of normal removal sounds way harder to me.

My experience if doing multiple sockets, removed the playfield and place upside down on horses so that the underside of the playfield is flat and facing up. Use a hot heavier duty soldering iron or a variable setting one on a pretty high temp. Since these are old parts it doesn't matter if you "overheat" them. Then you have to remove the old wires and solder the new lamps. With the new lamps, you don't need as hot an iron.

For me, having the playfield out of the game and flat makes it a lot easier to reach the lamps and I don't have to worry about solder splatter as much. Also, take pictures if you are doing a bunch just to make sure you put them back correctly. It's easy to accidentally switch lamp wires on lights that are close together to each other especially if you take a break that turns into a couple of days. You can come back and not be sure which wire is which without looking at the schematics.

Yep, then just high heat and a little electronic screwdriver my friend. As said above if you can take the PF out its much easier

You need a bigger iron.

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Make sure to use a jumper on the grounds to see how many are sockets and not issues with the wiring

What Grumpy said. It's not entirely about a hotter iron (of course, that does help), it's more about contact area. A chisel tip is perfect for this application. You can maybe put the tip on it's side. Might be awkward, but will help.

How many watts is your iron?

I usually get some fat solder wick and wick off as much as I can in short order so I don't damage the board.

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Unless the solder blob prevents you from getting the head of the driver on the screw the solder is not strong enough to stop you from removing the screw.

If you cant get on the screw just heat the solder up and you should be able to remove the screw.